What are the most important steps to take to get a lot of qualified traffic to a new website as quickly as possible?

This question needs a bit more detail to be answered specifically, but long story short you will need to pay for it. Based on your audience, goals and objectives, you can fine tune the targeting to ensure you are paying for qualified traffic.

You can also start cranking out and distributing content, but paying for the traffic is going to cover the 'quickly' part better than any other option.

Depends on a whole bunch of factors. Do you have money to spend on online ads? Do you have people in house to build growth features? How niche is your website? Are there existing distribution platforms you can launch on? The following strategies have worked for me well depending on the type of product:

1. Building viral growth features directly into my site to amplify traffic (lots of potential here depending on your product/site)
2. Spending on online ads on Google, Facebook, etc. (if you want this to be sustainable, lifetime value should be significantly more than cost of customer acquisition, but this might not matter in the short term)
3. Generating PR (hard and short-term but when it works, it works)
4. Striking partnerships with other high volume sites OR getting on a platform that gives you immediate organic distribution. If you can get promoted by the platform, that can REALLY help.
5. Content marketing, SEO, etc. are slower but can be important in the long run

I've typically found that one strategy works best for any particular site so it's important to iterate until you find the best one.

As the others have mentioned you can always pay for advertising but this is a tricky process, there must be a million companies willing to take your money for "traffic" but there is so much more that is involved.

Targeted Traffic is what you need. Who is your target market, how old are they, where are they located, how much is in their income, and so on.

If you know your target market you can be more specific in how you direct traffic towards them. Once you have all that then you can either find a reputable company and set your monthly marketing budget or you can go the social marketing.

This will depend if you are online or not, and what it is your services do. There are a number of ways to build up traffic to a website but there are only a few ways to build up effective reoccurring traffic that will grow your business and your brand.